Simplify and expand the following expression: $ \dfrac{3}{n + 5}+ \dfrac{3}{5n + 35}+ \dfrac{2n}{n^2 + 12n + 35} $
Solution: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor a $5$ out of denominator in the second term: $ \dfrac{3}{5n + 35} = \dfrac{3}{5(n + 7)}$ We can factor the quadratic in the third term: $ \dfrac{2n}{n^2 + 12n + 35} = \dfrac{2n}{(n + 5)(n + 7)}$ Now we have: $ \dfrac{3}{n + 5}+ \dfrac{3}{5(n + 7)}+ \dfrac{2n}{(n + 5)(n + 7)} $ The least common multiple of the denominators is: $ (n + 5)(n + 7)$ In order to get the first term over $(n + 5)(n + 7)$ , multiply by $\dfrac{5(n + 7)}{5(n + 7)}$ $ \dfrac{3}{n + 5} \times \dfrac{5(n + 7)}{5(n + 7)} = \dfrac{15(n + 7)}{(n + 5)(n + 7)} $ In order to get the second term over $(n + 5)(n + 7)$ , multiply by $\dfrac{n + 5}{n + 5}$ $ \dfrac{3}{5(n + 7)} \times \dfrac{n + 5}{n + 5} = \dfrac{3(n + 5)}{(n + 5)(n + 7)} $ In order to get the third term over $(n + 5)(n + 7)$ , multiply by $\dfrac{5}{5}$ $ \dfrac{2n}{(n + 5)(n + 7)} \times \dfrac{5}{5} = \dfrac{10n}{(n + 5)(n + 7)} $ Now we have: $ \dfrac{15(n + 7)}{(n + 5)(n + 7)} + \dfrac{3(n + 5)}{(n + 5)(n + 7)} + \dfrac{10n}{(n + 5)(n + 7)} $ $ = \dfrac{ 15(n + 7) + 3(n + 5) + 10n} {(n + 5)(n + 7)} $ Expand: $ = \dfrac{15n + 105 + 3n + 15 + 10n}{5n^2 + 60n + 175} $ $ = \dfrac{28n + 120}{5n^2 + 60n + 175}$